This was a more personal letter than 1 John and is specifically addressed, possibly to a church which he terms as ‘the elect lady', with its members as her children. Based on Revelation 19:7, this would see the church as the forthcoming bride of Christ. But this sits ill with the mention of ‘children', even if they are spiritual children.
More likely it may have been directed to a particular lady of influence and position, possibly with a church meeting in her large household who had communicated with him about problems that they were having with false teachers. It is possible that this finally resulted in the sending of 1 John to the churches of the area.
He terms himself ‘the elder'. This appears to have been a title by which the Apostles were known (1 Peter 5:1 - although not exclusively) as we discover from the writings of Papias. An elder was someone who was in authority, and highly respected. To open his letter with ‘the elder' may have indicated that he knew that all would know who he was because it was a title used uniquely of him when used in the singular without a name. This would point to the Apostle John.